section is on Optimization:
Find the point on the curve y = x^2 closest to the point (3, 4)
To find the point on the curve closest to a given point, we can use optimization techniques. In this case, we want to find the point on the curve y = x^2 that is closest to the point (3, 4).
To approach this problem, we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In our case, our two points are (x, x^2) on the curve and (3, 4).
So, the distance between these two points is:
Distance = sqrt((x - 3)^2 + (x^2 - 4)^2)
To find the point on the curve closest to (3, 4), we need to minimize the distance function. To do this, we can take the derivative of the distance function with respect to x and set it equal to zero:
d(Distance)/dx = 0
Let's find the derivative of the distance function:
d(Distance)/dx = d/dx(sqrt((x - 3)^2 + (x^2 - 4)^2))
To simplify the derivative, we can first square the expression inside the square root:
d(Distance)/dx = d/dx(sqrt((x - 3)^2 + (x^2 - 4)^2))
= d/dx(sqrt((x - 3)^2 + (x^4 - 8x^2 + 16)))
Using the chain rule, we can find the derivative of this expression:
d(Distance)/dx = (1/2) * (x - 3)^(-1/2) * (2(x - 3) + 2(x^4 - 8x^2 + 16)(4x^3 - 16x))
Simplifying the derivative:
0 = (x - 3) + (x^4 - 8x^2 + 16)(4x^3 - 16x)
To find the x-coordinate of the point on the curve closest to (3, 4), we need to solve this equation. However, this equation is quite complex and difficult to solve analytically.
Instead, we can use numerical methods or graphing tools to find an approximate solution. One possible approach is to plot the distance function for various values of x and look for the minimum value. We can use graphing software or online tools to graph the function and find the x-coordinate that corresponds to the minimum distance.
Once we have the x-coordinate, we can substitute it into the equation y = x^2 to find the y-coordinate of the point on the curve closest to (3, 4).